Rabbi Kaplan was born in the Bronx, New York City, to the Sefardi Recanati family of Salonika, Greece. He studied at Yeshiva Torah Vodaas and the Mir yeshiva in Brooklyn. Kaplan received semicha from some of Israel's foremost rabbinic authorities, including Rabbi Eliezer Yehudah Finkel. After his rabbinic ordination, he earned a master's degree in physics. As a graduate student, Kaplan was described in a scientific "Who's Who" as a promising young American physicist.

His major influence was Rabbi Zvi Aryeh Rosenfeld (1922—1978), who single-handedly introduced the teachings of Rebbe Nachman of Breslov to American shores beginning in the 1950s, inspiring many students at Brooklyn yeshivas, especially Torah Vodaas. Working together, Kaplan and Rosenfeld translated and annotated Rabbi Nachman's Tikkun (based on the Tikkun HaKlali). At Rosenfeld's suggestion, Kaplan also produced the first-ever English translation of Sichot HaRan ("Rabbi Nachman's Wisdom"), which Rosenfeld edited. He also translated and annotated Until the Mashiach: The Life of Rabbi Nachman, a day-to-day account of Rebbe Nachman's life, for the newly-established Breslov Research Institute founded by Rosenfeld's son-in-law, Chaim Kramer. Kaplan's later writings further explored Hasidut, Kabbalah and Jewish meditation. (Kaplan himself utilized the meditative form of Kabbalah on a daily basis. )

From 1976 onward, Kaplan's major activity was the translation into English of the recently-translated (Ladino into Hebrew, 1967) anthology, Me'am Lo'ez. He also completed The Living Torah, a new translation of the Five Books of Moses and the Haftarot, shortly before his death. Kaplan was described by Rabbi Pinchas Stolper, his original sponsor, as never fearing to speak his mind. "He saw harmony between science and Judaism, where many others saw otherwise. He put forward creative and original ideas and hypotheses, all the time anchoring them in classical works of rabbinic literature." His works continue to attract a wide readership, and are studied by both novices and the newly religious, as well as by scholars where the extensive footnotes provide a unique resource.

He died suddenly of a heart attack on January 28, 1983, at the age of 48. He was buried on the Mount of Olives, off Aweiss street, in the part known as "Agudas Achim Anshei America" "Chelek Alef" (Portion 1).

Kaplan produced works on topics as varied as prayer, Jewish marriage and meditation; his writing was also remarkable in that it seamlessly incorporated ideas from across the spectrum of Rabbinic literature, including Kabbalah and Hasidut. His introductory and background material contain much scholarly and original research. In researching his books, Kaplan once remarked: "I use my physics background to analyze and systematize data, very much as a physicist would deal with physical reality." This ability enabled him to undertake monumental projects, producing close to 50 books. His works have been translated into Czech, French, Hungarian, Modern Hebrew, Portuguese, Russian, and Spanish.

• "The Living Torah", Rabbi Kaplan's best-known work, is a widely used, scholarly (and user friendly) translation into English of the Torah. It is noteworthy for its detailed index, thorough cross-references, extensive footnotes with maps and diagrams, and research on realia, flora, fauna, and geography. The footnotes also indicate differences in interpretation between the classic commentators. It was one of the first translations structured around the parshiyot, the traditional division of the Torah text. (Moznaim, 1981, ISBN 0-940118-35-1)

• "Handbook of Jewish Thought," produced early in his career, is an encyclopedic and systematic treatment of Judaism's fundamental beliefs. Because of the work's structure and detail, the references, with the index, can serve as a research resource across almost all of rabbinic literature. (Moznaim, Vol. 1, 1979, ISBN 0-940118-49-1; Vol. 2, 1992, ISBN 0-940118-79-3)

• "Torah Anthology," a 45-volume translation of Me'am Lo'ez from Ladino (Judæo-Spanish) into English. Rabbi Kaplan was the primary translator.

• "Tefillin: God, Man and Tefillin"; "Love Means Reaching Out"; "Maimonides' Principles"; "The Fundamentals of Jewish Faith"; "The Waters of Eden: The Mystery of the Mikvah"; "Jerusalem: Eye of the Universe" — a series of highly popular and influential booklets on aspects of Jewish philosophy which span the entire spectrum of Jewish thought, as well as various religious practices. Published by the Orthodox Union/NCSY. or as an anthology by Artscroll, 1991, ISBN 1-57819-468-7.

• Five booklets of the Young Israel Intercollegiate Hashkafa Series — "Belief in God"; "Free Will and the Purpose of Creation"; "The Jew"; "Love and the Commandments"; and "The Structure of Jewish Law" launched his writing career. He was also a frequent contributor to The Jewish Observer. (These articles have been published as a collection: Artscroll, 1986, ISBN 0-89906-173-7)

• "The Real Messiah? A Jewish Response to Missionaries"(PDF).

• Kaplan translated and annotated classic works on Jewish mysticism — "Sefer Yetzirah",, "Bahir", and "Derekh Hashem" — as well as produced much original work on the subject in English. His Moreh Ohr, a Hebrew-language work, discusses the purpose of Creation, tzimtzum and free will from a kabbalistic point of view.

• He wrote three well-known books on Jewish meditation. These works revive and reconstruct ancient Jewish practices and vocabulary relating to meditation.

• He wrote and translated several works related to Hasidic Judaism in general and to the teachings of Rabbi Nachman of Breslov in particular.